#set document(title: "3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem", author: "OpenStax / XYZ Homework")
#set page(width: 8.5in, height: auto, margin: 1in)
#import "@preview/cetz:0.5.2"
#set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en")
#show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math"))
#set par(justify: true, leading: 0.62em, spacing: 0.9em)
#set enum(spacing: 1.1em)   // room between list items so tall inline fractions don't collide
#set list(spacing: 1.1em)
#set table(stroke: 0.5pt + rgb("#c7ccd3"))

#let BLUE = rgb("#183B6F")     // brand navy — section bars + example/solution labels (white on navy 11.09:1)
#let ORANGE = rgb("#A94509")   // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text)
#let RED = rgb("#DC2626")      // brand error-600
#let GREEN = rgb("#059669")    // brand success-600 (decoration only; small green text uses green-text #007942)

#show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt,
  fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt,
  text(fill: white, weight: "bold", size: 19pt, it.body))
#show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt,
  fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt,
  text(fill: white, weight: "bold", size: 12pt, it.body))
#show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body)
#show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body)

#let examplebox(label, title, body) = block(width: 100%, breakable: true,
  fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt,
  above: 12pt, below: 12pt)[
  #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt,
    text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]]
  #body]

// rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade
#let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint,
    stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt),
    above: 11pt, below: 11pt)[
    #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)]
    #linebreak() #body]

#let solutionbox(body) = block(above: 4pt, below: 8pt)[
  #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body]

#let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"),
  stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[
  #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))]

// Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a
// consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the
// print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a
// degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero.
#let _STD_W = 3.5
#let _WIDE_W = 5.6
#let _MAX_H = 3.4
#let _ASPECT_WIDE = 2.2
#let _UPSCALE_MAX = 1.15
#let stdfig(body) = context {
  let m = measure(body)
  let w = calc.max(m.width / 1in, 0.01)
  let h = calc.max(m.height / 1in, 0.01)
  let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W }
  let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX)
  align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body)))
}

#show figure: set block(breakable: false)
#set figure(gap: 8pt)
#show figure.caption: set text(size: 8.5pt, fill: rgb("#555"))


== 3.4#h(0.6em)Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Simplify: #math.equation(block: false, alt: "the fraction 1 over 2 open parenthesis 6 h close parenthesis .")[$frac(1, 2) ( 6 h ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "3 h")[$3 h$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The length of a rectangle is three less than the width. Let #emph[w] represent the width. Write an expression for the length of the rectangle. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "l equals w minus 3")[$l = w − 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] for #emph[b] when #math.equation(block: false, alt: "A equals 260")[$A = 260$] and #math.equation(block: false, alt: "h equals 52 .")[$h = 52 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "b equals 10")[$b = 10$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 144 .")[$sqrt(144) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "12")[$12$]
]
]


=== Solve Applications Using Properties of Triangles

In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve Geometry Applications.]

+ #strong[Read] the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
+ #strong[Identify] what we are looking for.
+ #strong[Label] what we are looking for by choosing a variable to represent it.
+ #strong[Translate] into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer by substituting it back into the equation solved in step 5 and by making sure it makes sense in the context of the problem.
+ #strong[Answer] the question with a complete sentence.
]

We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

The plural of the word #emph[vertex] is #emph[vertices]. All triangles have three #strong[vertices]. Triangles are named by their vertices: The triangle in  is called #math.equation(block: false, alt: "△ A B C .")[$"△" A B C .$]

#figure(figph[A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively.], alt: "A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively.", caption: [Triangle ABC has vertices A, B, and C. The lengths of the sides are a, b, and c.])

The three angles of a triangle are related in a special way. The sum of their measures is #math.equation(block: false, alt: "180 ° .")[$180 "°" .$] Note that we read #math.equation(block: false, alt: "m ∠ A")[$m "∠" A$] as “the measure of angle A.” So in #math.equation(block: false, alt: "△ A B C")[$"△" A B C$] in ,

#math.equation(block: true, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180 °")[$m "∠" A + m "∠" B + m "∠" C = 180 "°"$]Because the #strong[perimeter] of a figure is the length of its boundary, the perimeter of #math.equation(block: false, alt: "△ A B C")[$"△" A B C$] is the sum of the lengths of its three sides.

#math.equation(block: true, alt: "P equals a plus b plus c")[$P = a + b + c$]To find the #strong[area] of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a #math.equation(block: false, alt: "90 °")[$90 "°"$] angle with the base. We will draw #math.equation(block: false, alt: "△ A B C")[$"△" A B C$] again, and now show the height, #emph[h].

#figure(figph[A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively. The side b is parallel to the bottom of the page, and it has a dashed line drawn from vertex B to it. This line is marked h and makes a right angle with side b.], alt: "A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively. The side b is parallel to the bottom of the page, and it has a dashed line drawn from vertex B to it. This line is marked h and makes a right angle with side b.", caption: [The formula for the area of #math.equation(block: false, alt: "△ A B C")[$"△" A B C$] is #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h ,")[$A = frac(1, 2) b h ,$] where #emph[b] is the base and #emph[h] is the height.])

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Triangle Properties]

#figure(figph[A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively. The side b is parallel to the bottom of the page, and it has a dashed line drawn from vertex B to it. This line is marked h and makes a right angle with side b.], alt: "A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively. The side b is parallel to the bottom of the page, and it has a dashed line drawn from vertex B to it. This line is marked h and makes a right angle with side b.", caption: none)

For #math.equation(block: false, alt: "△ A B C")[$"△" A B C$]

#strong[Angle measures:]

#math.equation(block: true, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180")[$m "∠" A + m "∠" B + m "∠" C = 180$]- The sum of the measures of the angles of a triangle is #math.equation(block: false, alt: "180 ° .")[$180 "°" .$]

#strong[Perimeter:]

#math.equation(block: true, alt: "P equals a plus b plus c")[$P = a + b + c$]- The perimeter is the sum of the lengths of the sides of the triangle.

#strong[Area:]

#math.equation(block: true, alt: "A equals the fraction 1 over 2 b h , b equals base , h equals height")[$A = frac(1, 2) b h , b = "base" , h = "height"$]- The area of a triangle is one-half the base times the height.
]

#examplebox("Example 1")[][
The measures of two angles of a triangle are 55 and 82 degrees. Find the measure of the third angle.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A triangle ABC is depicted with angle A measuring 82 degrees, angle B measuring 55 degrees, and angle C labeled as 'x'. The sum of angles in a triangle is 180 degrees.], alt: "A triangle ABC is depicted with angle A measuring 82 degrees, angle B measuring 55 degrees, and angle C labeled as 'x'. The sum of angles in a triangle is 180 degrees.", caption: none)]),
  [#emph[Step 2. Identify] what you are looking for.], [the measure of the third angle in a triangle],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [Let #math.equation(block: false, alt: "x equals")[$x =$] the measure of the angle.],
  [#emph[Step 4. Translate.]], [],
  [Write the appropriate formula and substitute.], [#math.equation(block: false, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180")[$m ∠ A + m ∠ B + m ∠ C = 180$]],
  [#emph[Step 5. Solve]the equation.], [#math.equation(block: false, alt: "55 plus 82 plus x, equals, 180; 137 plus x, equals, 180; x, equals, 43")[$55 + 82 + x & = & 180 \ 137 + x & = & 180 \ x & = & 43$]],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #math.equation(block: false, alt: "55 plus 82 plus 43, ≟, 180; 180, equals, 180 ✓")[$55 + 82 + 43 & ≟ & 180 \ 180 & = & 180 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The measure of the third angle is 43 degrees.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The measures of two angles of a triangle are 31 and 128 degrees. Find the measure of the third angle.

#solutionbox[
21 degrees
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The measures of two angles of a triangle are 49 and 75 degrees. Find the measure of the third angle.

#solutionbox[
56 degrees
]
]

#examplebox("Example 2")[][
The perimeter of a triangular garden is 24 feet. The lengths of two sides are four feet and nine feet. How long is the third side?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A triangle with sides labeled 4 ft, 9 ft, and c.], alt: "A triangle with sides labeled 4 ft, 9 ft, and c.", caption: none)

 #linebreak() #figure(figph[The image displays text on a white background, which reads 'P = 24 ft' in black letters, indicating a perimeter measurement of 24 feet.], alt: "The image displays text on a white background, which reads 'P = 24 ft' in black letters, indicating a perimeter measurement of 24 feet.", caption: none)]),
  [#emph[Step 2. Identify] what you are looking for.], [length of the third side of a triangle],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [Let #math.equation(block: false, alt: "c equals")[$c =$] the third side.],
  [#emph[Step 4. Translate.]], [],
  [Write the appropriate formula and substitute.], [#figure(figph[A mathematical equation displays P = a + b + c, with light blue braces positioned under each variable.], alt: "A mathematical equation displays P = a + b + c, with light blue braces positioned under each variable.", caption: none)],
  [Substitute in the given information.], [#figure(figph[A mathematical equation is displayed, reading '24 ft = 4 ft + 9 ft + C' against a white background.], alt: "A mathematical equation is displayed, reading '24 ft = 4 ft + 9 ft + C' against a white background.", caption: none)],
  [#emph[Step 5. Solve]the equation.], [#figure(figph[A mathematical equation is displayed, showing '24 = 13 + c'. This equation requires solving for the variable 'c'.], alt: "A mathematical equation is displayed, showing '24 = 13 + c'. This equation requires solving for the variable 'c'.", caption: none)

 #linebreak() #figure(figph[The image shows the handwritten equation '11 = c' on a plain white background.], alt: "The image shows the handwritten equation '11 = c' on a plain white background.", caption: none)],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #math.equation(block: false, alt: "P, equals, a plus b plus c; 24, ≟, 4 plus 9 plus 11; 24, equals, 24 ✓")[$P & = & a + b + c \ 24 & ≟ & 4 + 9 + 11 \ 24 & = & 24 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The third side is 11 feet long.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 feet. How long is the third side?

#solutionbox[
8 feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The lengths of two sides of a triangular window are seven feet and five feet. The perimeter is 18 feet. How long is the third side?

#solutionbox[
6 feet
]
]

#examplebox("Example 3")[][
The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[An isosceles triangle is shown with a base length of 15 m. A dashed line represents the height 'h' from the apex to the base, forming a right angle.], alt: "An isosceles triangle is shown with a base length of 15 m. A dashed line represents the height 'h' from the apex to the base, forming a right angle.", caption: none)

 #linebreak() Area #math.equation(block: false, alt: "equals 90 m squared")[$= 90 m^(2)$]]),
  [#emph[Step 2. Identify] what you are looking for.], [height of a triangle],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [Let #math.equation(block: false, alt: "h equals")[$h =$] the height.],
  [#emph[Step 4. Translate.]], [],
  [Write the appropriate formula.], [#figure(figph[The formula for calculating the area of a triangle, A = 1/2 \* b \* h, where A is the area, b is the base, and h is the height. Each component of the formula is highlighted with a blue bracket.], alt: "The formula for calculating the area of a triangle, A = 1/2 * b * h, where A is the area, b is the base, and h is the height. Each component of the formula is highlighted with a blue bracket.", caption: none)],
  [Substitute in the given information.], [#figure(figph[A mathematical equation is shown, displaying 90 m² = 1/2 ⋅ 15 m ⋅ h, likely representing the formula for the area of a triangle with known area and base, solving for height.], alt: "A mathematical equation is shown, displaying 90 m² = 1/2 ⋅ 15 m ⋅ h, likely representing the formula for the area of a triangle with known area and base, solving for height.", caption: none)],
  [#emph[Step 5. Solve]the equation.], [#figure(figph[A mathematical equation is displayed on a white background, showing 90 equals 15 over 2 multiplied by h.], alt: "A mathematical equation is displayed on a white background, showing 90 equals 15 over 2 multiplied by h.", caption: none)

 #linebreak() #figure(figph[The image displays a simple mathematical equation, '12 = h,' written in black text against a plain white background. The equation indicates that the value of 'h' is equal to 12.], alt: "The image displays a simple mathematical equation, '12 = h,' written in black text against a plain white background. The equation indicates that the value of 'h' is equal to 12.", caption: none)],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #math.equation(block: false, alt: "A, equals, the fraction 1 over 2 b h; 90, ≟, the fraction 1 over 2 times 15 times 12; 90, equals, 90 ✓")[$A & = & frac(1, 2) b h \ 90 & ≟ & frac(1, 2) ⋅ 15 ⋅ 12 \ 90 & = & 90 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The height of the triangle is 12 meters.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The area of a triangular painting is 126 square inches. The base is 18 inches. What is the height?

#solutionbox[
14 inches
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A triangular tent door has an area of 15 square feet. The height is five feet. What is the base?

#solutionbox[
6 feet
]
]

The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A #strong[right triangle] has one #math.equation(block: false, alt: "90 °")[$90 "°"$] angle, which we usually mark with a small square in the corner.

#figure(figph[A right triangle with the largest angle marked 90 degrees.], alt: "A right triangle with the largest angle marked 90 degrees.", caption: none)

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Right Triangle]

A #strong[right triangle] has one #math.equation(block: false, alt: "90 °")[$90 "°"$] angle, which is often marked with a square at the vertex.
]

#examplebox("Example 4")[][
One angle of a right triangle measures #math.equation(block: false, alt: "28 ° .")[$28 "°" .$] What is the measure of the third angle?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A right-angled triangle ABC with angle A = 90 degrees, angle B = 28 degrees, and angle C labeled as x.], alt: "A right-angled triangle ABC with angle A = 90 degrees, angle B = 28 degrees, and angle C labeled as x.", caption: none)]),
  [#emph[Step 2. Identify] what you are looking for.], [the measure of an angle],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [Let #math.equation(block: false, alt: "x equals")[$x =$] the measure of an angle.],
  [#emph[Step 4. Translate.]], [#math.equation(block: false, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180")[$m ∠ A + m ∠ B + m ∠ C = 180$]],
  [Write the appropriate formula and substitute.], [#math.equation(block: false, alt: "x plus 90 plus 28 equals 180")[$x + 90 + 28 = 180$]],
  [#emph[Step 5. Solve]the equation.], [#math.equation(block: false, alt: "x plus 118, equals, 180; x, equals, 62")[$x + 118 & = & 180 \ x & = & 62$]],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #math.equation(block: false, alt: "180, ≟, 90 plus 28 plus 62; 180, equals, 180 ✓")[$180 & ≟ & 90 + 28 + 62 \ 180 & = & 180 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The measure of the third angle is 62#super[°].],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One angle of a right triangle measures #math.equation(block: false, alt: "56 ° .")[$56 "°" .$] What is the measure of the other small angle?

#solutionbox[
#math.equation(block: true, alt: "34 °")[$34 "°"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One angle of a right triangle measures #math.equation(block: false, alt: "45 ° .")[$45 "°" .$] What is the measure of the other small angle?

#solutionbox[
#math.equation(block: true, alt: "45 °")[$45 "°"$]
]
]

In the examples we have seen so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.

#examplebox("Example 5")[][
The measure of one angle of a right triangle is 20 degrees more than the measure of the smallest angle. Find the measures of all three angles.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what you are looking for.], [the measures of all three angles],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [$#h(0.24em)$Let #math.equation(block: false, alt: "a equals 1 to the power st")[$a = 1^("st")$] angle. #linebreak() #math.equation(block: false, alt: "a plus 20 equals 2 to the power nd")[$a + 20 = 2^("nd")$] angle #linebreak() #math.equation(block: false, alt: "90 equals 3 to the power rd")[$#h(1.7em) 90 = 3^("rd")$] angle (the right angle)],
  [Draw the figure and label it with the given information], [#figure(figph[A right-angled triangle ABC, with angle C at 90 degrees, angle A labeled 'a', and angle B labeled 'a + 20'.], alt: "A right-angled triangle ABC, with angle C at 90 degrees, angle A labeled 'a', and angle B labeled 'a + 20'.", caption: none)],
  [#emph[Step 4. Translate]], [#figure(figph[This equation, m∠A + m∠B + m∠C = 180, represents the angle sum property of a triangle, where the sum of its interior angles is 180 degrees.], alt: "This equation, m∠A + m∠B + m∠C = 180, represents the angle sum property of a triangle, where the sum of its interior angles is 180 degrees.", caption: none)],
  [Write the appropriate formula. #linebreak() Substitute into the formula.], [#figure(figph[A mathematical equation is displayed: a + (a + 20) + 90 = 180. This equation likely represents a geometry problem, possibly related to angles in a triangle or on a straight line, where 'a' is an unknown value to be solved for.], alt: "A mathematical equation is displayed: a + (a + 20) + 90 = 180. This equation likely represents a geometry problem, possibly related to angles in a triangle or on a straight line, where 'a' is an unknown value to be solved for.", caption: none)],
  [#emph[Step 5. Solve]the equation.], [#figure(figph[A mathematical equation is displayed with a grey gradient effect against a white background, reading '2a + 110 = 180'.], alt: "A mathematical equation is displayed with a grey gradient effect against a white background, reading '2a + 110 = 180'.", caption: none)

 #linebreak() #figure(figph[A mathematical equation,], alt: "A mathematical equation,", caption: none)

 #linebreak() #figure(figph[The text 'a = 35 first angle' is displayed on a white background.], alt: "The text 'a = 35 first angle' is displayed on a white background.", caption: none)

 #linebreak() #figure(figph[The image displays the text 'a + 20 second angle' on a white background.], alt: "The image displays the text 'a + 20 second angle' on a white background.", caption: none)

 #linebreak() #figure(figph[The numbers 35 and 20 are displayed in white, with 35 in a slightly redder hue, separated by a plus sign, all against a clean white background, indicating an addition problem.], alt: "The numbers 35 and 20 are displayed in white, with 35 in a slightly redder hue, separated by a plus sign, all against a clean white background, indicating an addition problem.", caption: none)

 #linebreak() 55 #linebreak()          90 third angle],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #math.equation(block: false, alt: "35 plus 55 plus 90, ≟, 180; 180, equals, 180 ✓")[$35 + 55 + 90 & ≟ & 180 \ 180 & = & 180 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The three angles measure 35#super[°], 55#super[°], and 90#super[°].],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The measure of one angle of a right triangle is 50° more than the measure of the smallest angle. Find the measures of all three angles.

#solutionbox[
#math.equation(block: true, alt: "20 ° , 70 ° , 90 °")[$20 "°" , 70 "°" , 90 "°"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The measure of one angle of a right triangle is 30° more than the measure of the smallest angle. Find the measures of all three angles.

#solutionbox[
#math.equation(block: true, alt: "30 ° , 60 ° , 90 °")[$30 "°" , 60 "°" , 90 "°"$]
]
]


=== Use the Pythagorean Theorem

We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the #strong[Pythagorean Theorem]. This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.

Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a #math.equation(block: false, alt: "90 °")[$90 "°"$] angle, marked with a small square in the corner. The side of the triangle opposite the #math.equation(block: false, alt: "90 °")[$90 "°"$] angle is called the #emph[hypotenuse] and each of the other sides are called #emph[legs].

#figure(figph[Three right triangles with different orientations. The right angles are marked with two small lines that make a small square with the angle. Opposite these angles, hypotenuse is written. The other sides are marked “leg.”], alt: "Three right triangles with different orientations. The right angles are marked with two small lines that make a small square with the angle. Opposite these angles, hypotenuse is written. The other sides are marked “leg.”", caption: none)

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle, #math.equation(block: false, alt: "a squared plus b squared equals c squared ,")[$a^(2) + b^(2) = c^(2) ,$] where #math.equation(block: false, alt: "a and b")[$a #h(0.2em) "and" #h(0.2em) b$] are the lengths of the legs and #math.equation(block: false, alt: "c")[$c$] is the length of the hypotenuse.

Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[The Pythagorean Theorem]

In any right triangle, #math.equation(block: false, alt: "a squared plus b squared equals c squared .")[$a^(2) + b^(2) = c^(2) .$]

#figure(figph[A right triangle with sides marked a, b, and c. The side marked c is the hypotenuse.], alt: "A right triangle with sides marked a, b, and c. The side marked c is the hypotenuse.", caption: none)

where #emph[a] and #emph[b] are the lengths of the legs, #emph[c] is the length of the hypotenuse.
]

To solve exercises that use the Pythagorean Theorem, we will need to find square roots. We have used the notation #math.equation(block: false, alt: "the square root of m")[$sqrt(m)$] and the definition:

If #math.equation(block: false, alt: "m equals n squared ,")[$m = n^(2) ,$] then #math.equation(block: false, alt: "the square root of m equals n ,")[$sqrt(m) = n ,$] for #math.equation(block: false, alt: "n greater than or equal to 0 .")[$n ≥ 0 .$]

For example, we found that #math.equation(block: false, alt: "the square root of 25")[$sqrt(25)$] is 5 because #math.equation(block: false, alt: "25 equals 5 squared .")[$25 = 5^(2) .$]

Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

#examplebox("Example 6")[][
Use the Pythagorean Theorem to find the length of the hypotenuse shown below.

#figure(figph[A right triangle with legs marked 3 and 4.], alt: "A right triangle with legs marked 3 and 4.", caption: none)

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what you are looking for.], [the length of the hypotenuse of the triangle],
  [#emph[Step 3. Name.] Choose a variable to represent it. #linebreak() Label side #emph[c] on the figure.], [Let #emph[c] = the length of the hypotenuse. #linebreak()  #linebreak() #figure(figph[A right triangle diagram illustrating the Pythagorean theorem with legs 3 and 4, and hypotenuse 'c'. This setup is famously known as a 3-4-5 triangle.], alt: "A right triangle diagram illustrating the Pythagorean theorem with legs 3 and 4, and hypotenuse 'c'. This setup is famously known as a 3-4-5 triangle.", caption: none)],
  [#emph[Step 4. Translate.]], [],
  [Write the appropriate formula.], [#math.equation(block: false, alt: "a squared plus b squared equals c squared")[$a^(2) + b^(2) = c^(2)$]],
  [Substitute.], [#math.equation(block: false, alt: "3 squared plus 4 squared equals c squared")[$3^(2) + 4^(2) = c^(2)$]],
  [#emph[Step 5. Solve]the equation.], [#math.equation(block: false, alt: "9 plus 16 equals c squared")[$#h(0.4em) 9 + 16 = c^(2)$]],
  [Simplify.], [#math.equation(block: false, alt: "25 equals c squared")[$#h(2em) 25 = c^(2)$]],
  [Use the definition of square root.], [#math.equation(block: false, alt: "the square root of 25 equals c")[$#h(1.5em) sqrt(25) = c$]],
  [Simplify.], [#math.equation(block: false, alt: "5 equals c")[$#h(2.5em) 5 = c$]],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #figure(figph[A step-by-step verification of the Pythagorean theorem for a 3-4-5 right triangle, showing 3^2 + 4^2 = 5^2 simplifies to 25 = 25.], alt: "A step-by-step verification of the Pythagorean theorem for a 3-4-5 right triangle, showing 3^2 + 4^2 = 5^2 simplifies to 25 = 25.", caption: none)], [],
  [#emph[Step 7. Answer]the question.], [The length of the hypotenuse is 5.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

#figure(figph[A right triangle with legs marked 6 and 8. The hypotenuse is marked c.], alt: "A right triangle with legs marked 6 and 8. The hypotenuse is marked c.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "c equals 10")[$c = 10$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

#figure(figph[The image shows a right triangle. The number 5 is written on the base and the number 12 is written on the vertical leg. The hypotenuse is labeled c.], alt: "The image shows a right triangle. The number 5 is written on the base and the number 12 is written on the vertical leg. The hypotenuse is labeled c.", caption: none)

#solutionbox[
#math.equation(block: true, alt: "c equals 13")[$c = 13$]
]
]

#examplebox("Example 7")[][
Use the Pythagorean Theorem to find the length of the leg shown below.

#figure(figph[A right angle with one leg marked 5. The hypotenuse is labeled 13.], alt: "A right angle with one leg marked 5. The hypotenuse is labeled 13.", caption: none)

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what you are looking for.], [], [the length of the leg of the triangle],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [], [Let #emph[b] = the leg of the triangle.],
  [Lable side #emph[b].], [], [#figure(figph[A right-angled triangle with one leg of length 5, the hypotenuse of length 13, and the other leg labeled 'b'. The right angle is shown at the top-left vertex.], alt: "A right-angled triangle with one leg of length 5, the hypotenuse of length 13, and the other leg labeled 'b'. The right angle is shown at the top-left vertex.", caption: none)],
  [#emph[Step 4. Translate]], [], [],
  [Write the appropriate formula.], [], [#math.equation(block: false, alt: "a squared plus b squared equals c squared")[$a^(2) + b^(2) = c^(2)$]],
  [Substitute.], [], [#math.equation(block: false, alt: "5 squared plus b squared equals 13 squared")[$5^(2) + b^(2) = 13^(2)$]],
  [#emph[Step 5. Solve]the equation.], [], [#math.equation(block: false, alt: "25 plus b squared equals 169")[$25 + b^(2) = 169$]],
  [Isolate the variable term.], [], [#math.equation(block: false, alt: "b squared equals 144")[$#h(2.1em) b^(2) = 144$]],
  [Use the definition of square root.], [], [#math.equation(block: false, alt: "b squared equals the square root of 144")[$#h(2.1em) b^(2) = sqrt(144)$]],
  [Simplify.], [], [#math.equation(block: false, alt: "b equals 12")[$#h(2.6em) b = 12$]],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #figure(figph[Step-by-step math showing 5 squared plus 12 squared equals 13 squared. The calculation 25 + 144 = 169 confirms this classic Pythagorean triple.], alt: "Step-by-step math showing 5 squared plus 12 squared equals 13 squared. The calculation 25 + 144 = 169 confirms this classic Pythagorean triple.", caption: none)], [], [],
  [#emph[Step 7. Answer]the question.], [], [The length of the leg is 12.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

#figure(figph[A right triangle with legs marked b and 15. The hypotenuse is marked 17.], alt: "A right triangle with legs marked b and 15. The hypotenuse is marked 17.", caption: none)

#solutionbox[
8
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

#figure(figph[A right triangle with legs marked b and 9. The hypotenuse is marked 15.], alt: "A right triangle with legs marked b and 9. The hypotenuse is marked 15.", caption: none)

#solutionbox[
12
]
]

#examplebox("Example 8")[][
#figure(figph[A gazebo is shown. In one of its corners, a triangle is made with the wood. The hypotenuse is marked 10 inches, and one of the legs is marked x], alt: "A gazebo is shown. In one of its corners, a triangle is made with the wood. The hypotenuse is marked 10 inches, and one of the legs is marked x", caption: none)

Kelvin is building a gazebo and wants to brace each corner by placing a #math.equation(block: false, alt: "10 ″")[$10 "″"$] piece of wood diagonally as shown above.

If he fastens the wood so that the ends of the brace are the same distance from the corner, what is the length of the legs of the right triangle formed? Approximate to the nearest tenth of an inch.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what we are looking for.], [the distance from the corner that the bracket should be attached],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [Let #math.equation(block: false, alt: "x equals")[$x =$] the distance from the corner.],
  [#emph[Step 4. Translate] #linebreak() Write the appropriate formula and substitute.], [#math.equation(block: false, alt: "a squared plus b squared, equals, c squared; x squared plus x squared, equals, 10 squared")[$a^(2) + b^(2) & = & c^(2) \ x^(2) + x^(2) & = & 10^(2)$]],
  [#emph[Step 5. Solve] the equation.
		 #linebreak() Isolate the variable.
		 #linebreak() Use the definition of square root.
		 #linebreak() Simplify. Approximate to the nearest tenth.], [#math.equation(block: false, alt: "2 x squared, equals, 100; x squared, equals, 50; x, equals, the square root of 50; x, approximately equals, 7.1")[$2 x^(2) & = & 100 \ x^(2) & = & 50 \ x & = & sqrt(50) \ x & ≈ & 7.1$]],
  [#emph[Step 6. Check.]
 		 #linebreak() #math.equation(block: false, alt: "a squared plus b squared, equals, c squared; open parenthesis 7.1 close parenthesis squared plus open parenthesis 7.1 close parenthesis squared, approximately equals, 10 squared Yes.")[$#h(2.5em) a^(2) + b^(2) & = & c^(2) \ #h(2.5em) attach(( 7.1 ), t: 2) + attach(( 7.1 ), t: 2) & ≈ & 10^(2) #h(0.2em) "Yes."$]], [],
  [#emph[Step 7. Answer] the question.], [Kelvin should fasten each piece of wood approximately 7.1" from the corner.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. How far up the wall does the ladder reach?

#figure(figph[A house is shown with a ladder leaning against it. The ladder is marked 13’, and the distance from the house to the base of the ladder is marked 5’.], alt: "A house is shown with a ladder leaning against it. The ladder is marked 13’, and the distance from the house to the base of the ladder is marked 5’.", caption: none)

#solutionbox[
12 feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Randy wants to attach a 17 foot string of lights to the top of the 15 foot mast of his sailboat, as shown below. How far from the base of the mast should he attach the end of the light string?

#figure(figph[A sailboat is shown with a 15’ mast (the straight tall part). From the top of the mast, a series of colored dots stretches down to the back of the boat and is marked 17’.], alt: "A sailboat is shown with a 15’ mast (the straight tall part). From the top of the mast, a series of colored dots stretches down to the back of the boat and is marked 17’.", caption: none)

#solutionbox[
8 feet
]
]


=== Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right #math.equation(block: false, alt: "open parenthesis 90 ° close parenthesis")[$( 90 "°" )$] angles. The opposite sides of a #strong[rectangle] are the same length. We refer to one side of the rectangle as the length, #emph[L], and its adjacent side as the width, #emph[W].

#figure(figph[A rectangle with sides marked W and L.], alt: "A rectangle with sides marked W and L.", caption: none)

The distance around this rectangle is #math.equation(block: false, alt: "L plus W plus L plus W ,")[$L + W + L + W ,$] or #math.equation(block: false, alt: "2 L plus 2 W .")[$2 L + 2 W .$] This is the #strong[perimeter], #emph[P], of the rectangle.

#math.equation(block: true, alt: "P equals 2 L plus 2 W")[$P = 2 L + 2 W$]What about the #strong[area] of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.

#figure(figph[A rectangles composed of 6 squares that is three high and two wide. The height is marked 3 and the width is marked 2.], alt: "A rectangles composed of 6 squares that is three high and two wide. The height is marked 3 and the width is marked 2.", caption: none)

#math.equation(block: true, alt: "A equals 6; A equals 2 times 3; A equals L times W")[$A = 6 \ A = 2 · 3 \ A = L · W$]The area is the length times the width.

The formula for the area of a rectangle is #math.equation(block: false, alt: "A equals L W .")[$A = L W .$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of Rectangles]

Rectangles have four sides and four right #math.equation(block: false, alt: "open parenthesis 90 ° close parenthesis")[$( 90 "°" )$] angles.

The lengths of opposite sides are equal.

The perimeter of a rectangle is the sum of twice the length and twice the width.

#math.equation(block: true, alt: "P equals 2 L plus 2 W")[$P = 2 L + 2 W$]The area of a rectangle is the product of the length and the width.

#math.equation(block: true, alt: "A equals L times W")[$A = L · W$]
]

#examplebox("Example 9")[][
The length of a rectangle is 32 meters and the width is 20 meters. What is the perimeter?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. #linebreak() Draw the figure and label it with the given information.], [#figure(figph[A clear, simple line drawing of a rectangle. The top and bottom sides are labeled '32 m', and the left and right sides are labeled '20 m', indicating its dimensions.], alt: "A clear, simple line drawing of a rectangle. The top and bottom sides are labeled '32 m', and the left and right sides are labeled '20 m', indicating its dimensions.", caption: none)]),
  [#emph[Step 2. Identify] what you are looking for.], [the perimeter of a rectangle],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [Let #emph[P =] the perimeter.],
  [#emph[Step 4. Translate].], [],
  [Write the appropriate formula.], [#figure(figph[The formula P = 2L + 2W is displayed, representing the perimeter of a rectangle where P is perimeter, L is length, and W is width. Each component of the equation is bracketed below with light blue braces.], alt: "The formula P = 2L + 2W is displayed, representing the perimeter of a rectangle where P is perimeter, L is length, and W is width. Each component of the equation is bracketed below with light blue braces.", caption: none)],
  [Substitute.], [#figure(figph[A mathematical equation is displayed on a white background: p = 2(32 m) + 2(20 m). The equation shows the calculation of a perimeter 'p' using two sets of measurements, 32 meters and 20 meters, each multiplied by 2 and then added together.], alt: "A mathematical equation is displayed on a white background: p = 2(32 m) + 2(20 m). The equation shows the calculation of a perimeter 'p' using two sets of measurements, 32 meters and 20 meters, each multiplied by 2 and then added together.", caption: none)],
  [#emph[Step 5. Solve]the equation.], [#figure(figph[The image displays a mathematical equation, 'P = 64 + 40,' written in a dark gray font against a plain white background.], alt: "The image displays a mathematical equation, 'P = 64 + 40,' written in a dark gray font against a plain white background.", caption: none)

 #linebreak() #figure(figph[The image displays the text 'P = 104' in a grayscale font on a plain white background.], alt: "The image displays the text 'P = 104' in a grayscale font on a plain white background.", caption: none)],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #math.equation(block: false, alt: "P, ≟, 104; 20 plus 32 plus 20 plus 32, ≟, 104; 104, equals, 104 ✓")[$P & ≟ & 104 \ 20 + 32 + 20 + 32 & ≟ & 104 \ 104 & = & 104 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The perimeter of the rectangle is 104 meters.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The length of a rectangle is 120 yards and the width is 50 yards. What is the perimeter?

#solutionbox[
340 yards
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The length of a rectangle is 62 feet and the width is 48 feet. What is the perimeter?

#solutionbox[
220 feet
]
]

#examplebox("Example 10")[][
The area of a rectangular room is 168 square feet. The length is 14 feet. What is the width?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. #linebreak() Draw the figure and label it with the given information.], [#figure(figph[A rectangle is shown with one side labeled 'W' and an adjacent side labeled '14 ft'.], alt: "A rectangle is shown with one side labeled 'W' and an adjacent side labeled '14 ft'.", caption: none)]),
  [#emph[Step 2. Identify] what you are looking for.], [the width of a rectangular room],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [Let #emph[W] = the width.],
  [#emph[Step 4. Translate].], [],
  [Write the appropriate formula.], [#math.equation(block: false, alt: "A equals L W")[$#h(0.9em) A = L W$]],
  [Substitute.], [#math.equation(block: false, alt: "168 equals 14 W")[$#h(0.2em) 168 = 14 W$]],
  [#emph[Step 5. Solve]the equation.], [#math.equation(block: false, alt: "the fraction 168 over 14 equals the fraction 14 W over 14")[$frac(168, 14) = frac(14 W, 14)$] #linebreak() #math.equation(block: false, alt: "12 equals W")[$#h(0.6em) 12 = W$]],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #figure(figph[A rectangle with dimensions of 12 ft by 14 ft.], alt: "A rectangle with dimensions of 12 ft by 14 ft.", caption: none)

 #linebreak() #math.equation(block: false, alt: "A, equals, L W; 168, ≟, 14 times 12; 168, equals, 168 ✓")[$A & = & L W \ 168 & ≟ & 14 ⋅ 12 \ 168 & = & 168 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The width of the room is 12 feet.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The area of a rectangle is 598 square feet. The length is 23 feet. What is the width?

#solutionbox[
26 feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The width of a rectangle is 21 meters. The area is 609 square meters. What is the length?

#solutionbox[
29 meters
]
]

#examplebox("Example 11")[][
Find the length of a rectangle with perimeter 50 inches and width 10 inches.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. #linebreak() Draw the figure and label it with the given information.], [#figure(figph[A rectangle with a height of 10 inches and a base labeled L.], alt: "A rectangle with a height of 10 inches and a base labeled L.", caption: none)

 #linebreak() #figure(figph[The text 'P = 50 in' is displayed on a white background.], alt: "The text 'P = 50 in' is displayed on a white background.", caption: none)]),
  [#emph[Step 2. Identify] what you are looking for.], [the length of the rectangle],
  [#emph[Step 3. Name.] Choose a variable to represent it.], [Let #emph[L] = the length.],
  [#emph[Step 4. Translate].], [],
  [Write the appropriate formula.], [#figure(figph[The mathematical formula P = 2L + 2W, which is used to calculate the perimeter of a rectangle, is shown on a white background.], alt: "The mathematical formula P = 2L + 2W, which is used to calculate the perimeter of a rectangle, is shown on a white background.", caption: none)],
  [Substitute.], [#figure(figph[A mathematical equation is displayed, showing '50 = 2L + 2(10)' in black text against a white background.], alt: "A mathematical equation is displayed, showing '50 = 2L + 2(10)' in black text against a white background.", caption: none)],
  [#emph[Step 5. Solve]the equation.], [#figure(figph[A mathematical equation is displayed on a white background: 50 - 20 = 2L + 20 - 20. The numbers '20' in the subtraction and addition operations on both sides are highlighted in red.], alt: "A mathematical equation is displayed on a white background: 50 - 20 = 2L + 20 - 20. The numbers '20' in the subtraction and addition operations on both sides are highlighted in red.", caption: none)

 #linebreak()  #linebreak() #figure(figph[A mathematical expression '30 = 2L' displayed on a white background, representing an equation with a variable 'L'.], alt: "A mathematical expression '30 = 2L' displayed on a white background, representing an equation with a variable 'L'.", caption: none)

 #linebreak()  #linebreak() #figure(figph[The mathematical equation 30/2 = 2L/2 is shown on a white background, demonstrating a basic algebraic problem involving fractions.], alt: "The mathematical equation 30/2 = 2L/2 is shown on a white background, demonstrating a basic algebraic problem involving fractions.", caption: none)

 #linebreak()  #linebreak() #figure(figph[The image displays a white background with the equation '15 = L' written in black in the upper left corner.], alt: "The image displays a white background with the equation '15 = L' written in black in the upper left corner.", caption: none)],
  [#emph[Step 6. Check.] #linebreak() #figure(figph[A rectangle is shown with its sides labeled. The top and bottom sides are 15 inches long, and the left and right sides are 10 inches long.], alt: "A rectangle is shown with its sides labeled. The top and bottom sides are 15 inches long, and the left and right sides are 10 inches long.", caption: none)

 #linebreak() #math.equation(block: false, alt: "P, equals, 50; 15 plus 10 plus 15 plus 10, ≟, 50; 50, equals, 50 ✓")[$P & = & 50 \ 15 + 10 + 15 + 10 & ≟ & 50 \ 50 & = & 50 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The length is 15 inches.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the length of a rectangle with: perimeter 80 and width 25.

#solutionbox[
15
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the length of a rectangle with: perimeter 30 and width 6.

#solutionbox[
9
]
]

We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

#examplebox("Example 12")[][
The width of a rectangle is two feet less than the length. The perimeter is 52 feet. Find the length and width.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what you are looking for.], [the length and width of a rectangle],
  [#emph[Step 3. Name.] Choose a variable to represent it. #linebreak() Since the width is defined in terms of the length, we let #emph[L =]length. The width is two feet less than the length, so we let #emph[L] − 2 = width.], [#figure(figph[A simple geometric shape, a rectangle, is displayed with its bottom side labeled 'L' and its left side labeled 'L-2'. This indicates the algebraic expressions for its dimensions.], alt: "A simple geometric shape, a rectangle, is displayed with its bottom side labeled 'L' and its left side labeled 'L-2'. This indicates the algebraic expressions for its dimensions.", caption: none)

 #linebreak() #math.equation(block: false, alt: "P equals 52")[$#h(4em) P = 52$] ft],
  [#emph[Step 4. Translate].], [],
  [Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information.], [#math.equation(block: false, alt: "P equals 2 L plus 2 W")[$P = 2 L + 2 W$]],
  [Substitute in the given information.], [#math.equation(block: false, alt: "52 equals 2 L plus 2 open parenthesis L minus 2 close parenthesis")[$52 = 2 L + 2 ( L − 2 )$]],
  [#emph[Step 5. Solve]the equation.], [#math.equation(block: false, alt: "52 equals 2 L plus 2 L minus 4")[$52 = 2 L + 2 L − 4$]],
  [Combine like terms.], [#math.equation(block: false, alt: "52 equals 4 L minus 4")[$52 = 4 L − 4$]],
  [Add 4 to each side.], [#math.equation(block: false, alt: "56 equals 4 L")[$56 = 4 L$]],
  [Divide by 4.], [#math.equation(block: false, alt: "the fraction 56 over 4 equals the fraction 4 L over 4")[$frac(56, 4) = frac(4 L, 4)$] #linebreak() #math.equation(block: false, alt: "14 equals L")[$14 = L$] #linebreak() The length is 14 feet.],
  [Now we need to find the width.], [The width is #math.equation(block: false, alt: "L minus 2")[$L − 2$]. #linebreak() #figure(figph[Mathematical expressions on a white background, showing 'L - 2' followed by '14 - 2' with the number 14 in red, and then the result '12' below it.], alt: "Mathematical expressions on a white background, showing 'L - 2' followed by '14 - 2' with the number 14 in red, and then the result '12' below it.", caption: none)

 #linebreak() The width is 12 feet.],
  [#emph[Step 6. Check.] #linebreak() Since #math.equation(block: false, alt: "14 plus 12 plus 14 plus 12 equals 52")[$14 + 12 + 14 + 12 = 52$], this works!  #linebreak() #figure(figph[A rectangle with a length of 14 ft and a width of 12 ft.], alt: "A rectangle with a length of 14 ft and a width of 12 ft.", caption: none)

 #linebreak()], [],
  [#emph[Step 7. Answer]the question.], [The length is 14 feet and the width is 12 feet.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The width of a rectangle is seven meters less than the length. The perimeter is 58 meters. Find the length and width.

#solutionbox[
18 meters, 11 meters
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The length of a rectangle is eight feet more than the width. The perimeter is 60 feet. Find the length and width.

#solutionbox[
19 feet, 11 feet
]
]

#examplebox("Example 13")[][
The length of a rectangle is four centimeters more than twice the width. The perimeter is 32 centimeters. Find the length and width.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what you are looking for.], [the length and the width],
  [#emph[Step 3. Name.] Choose a variable to represent the width.], [#figure(figph[The text 'Let W = width' is displayed on a white background, likely defining a variable in a mathematical or scientific context.], alt: "The text 'Let W = width' is displayed on a white background, likely defining a variable in a mathematical or scientific context.", caption: none)],
  [The length is four more than twice the width.], [#figure(figph[A mathematical equation displayed on a white background, which reads '2W + 4 = length' in a dark gray font, representing a formula for calculating length based on a variable 'W'.], alt: "A mathematical equation displayed on a white background, which reads '2W + 4 = length' in a dark gray font, representing a formula for calculating length based on a variable 'W'.", caption: none)

 #linebreak()  #linebreak() #figure(figph[A rectangle is shown with its width labeled W and its length labeled 2W + 4.], alt: "A rectangle is shown with its width labeled W and its length labeled 2W + 4.", caption: none)

 #linebreak() #figure(figph[The image displays text indicating 'P = 32 cm' on a plain white background, likely representing a perimeter measurement in a mathematical or geometric context.], alt: "The image displays text indicating 'P = 32 cm' on a plain white background, likely representing a perimeter measurement in a mathematical or geometric context.", caption: none)],
  [#emph[Step 4. Translate]], [],
  [Write the appropriate formula.], [#figure(figph[A mathematical formula is displayed on a white background, reading 'P = 2L + 2W'. This represents the perimeter of a rectangle, where P is perimeter, L is length, and W is width.], alt: "A mathematical formula is displayed on a white background, reading 'P = 2L + 2W'. This represents the perimeter of a rectangle, where P is perimeter, L is length, and W is width.", caption: none)],
  [Substitute in the given information.], [#figure(figph[A mathematical equation is displayed, stating 32 equals 2 multiplied by the sum of 2W and 4, plus 2W. The equation is '32 = 2(2W + 4) + 2W', set against a plain white background.], alt: "A mathematical equation is displayed, stating 32 equals 2 multiplied by the sum of 2W and 4, plus 2W. The equation is '32 = 2(2W + 4) + 2W', set against a plain white background.", caption: none)],
  [#emph[Step 5. Solve]the equation.], [#figure(figph[A mathematical equation is displayed, showing '32 = 4W + 8 + 2W' against a white background.], alt: "A mathematical equation is displayed, showing '32 = 4W + 8 + 2W' against a white background.", caption: none)

 #linebreak() #figure(figph[A mathematical equation is displayed on a white background: 32 = 6W + 8.], alt: "A mathematical equation is displayed on a white background: 32 = 6W + 8.", caption: none)

 #linebreak() #figure(figph[A white background features the simple algebraic equation 24 = 6W written in black text. The image clearly displays the mathematical problem for solving W.], alt: "A white background features the simple algebraic equation 24 = 6W written in black text. The image clearly displays the mathematical problem for solving W.", caption: none)

 #linebreak() #figure(figph[The image displays mathematical notation stating that the number 4 is equal to W, which is further identified as 'width' in parentheses. The text is in a black font on a plain white background, indicating a definition or assignment of a variable.], alt: "The image displays mathematical notation stating that the number 4 is equal to W, which is further identified as 'width' in parentheses. The text is in a black font on a plain white background, indicating a definition or assignment of a variable.", caption: none)

 #linebreak() #figure(figph[A mathematical expression reads '2W + 4 (length)', likely representing a formula for calculating length based on a variable W.], alt: "A mathematical expression reads '2W + 4 (length)', likely representing a formula for calculating length based on a variable W.", caption: none)

 #linebreak()  #linebreak() #figure(figph[A mathematical expression reads 2(4) + 4, with the number 4 inside the parentheses highlighted in red.], alt: "A mathematical expression reads 2(4) + 4, with the number 4 inside the parentheses highlighted in red.", caption: none)

 #linebreak()    12 #linebreak() The length is 12 cm.],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #figure(figph[A rectangle with a height of 4 cm and a width of 12 cm.], alt: "A rectangle with a height of 4 cm and a width of 12 cm.", caption: none)

 #linebreak() #math.equation(block: false, alt: "P, equals, 2 L plus 2 W; 32, ≟, 2 times 12 plus 2 times 4; 32, equals, 32 ✓")[$P & = & 2 L + 2 W \ 32 & ≟ & 2 ⋅ 12 + 2 ⋅ 4 \ 32 & = & 32 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The length is 12 cm and the width is 4 cm.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The length of a rectangle is eight more than twice the width. The perimeter is 64. Find the length and width.

#solutionbox[
24, 8
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The width of a rectangle is six less than twice the length. The perimeter is 18. Find the length and width.

#solutionbox[
5, 4
]
]

#examplebox("Example 14")[][
The perimeter of a rectangular swimming pool is 150 feet. The length is 15 feet more than the width. Find the length and width.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. #linebreak() Draw the figure and label it with the given information.], [#figure(figph[A rectangular swimming pool with lanes, labeled with algebraic dimensions. The width is 'W' and the length is 'W + 15'.], alt: "A rectangular swimming pool with lanes, labeled with algebraic dimensions. The width is 'W' and the length is 'W + 15'.", caption: none)

 #linebreak() #math.equation(block: false, alt: "P equals 150")[$#h(3em) P = 150$] ft]),
  [#emph[Step 2. Identify] what you are looking for.], [the length and the width of the pool],
  [#emph[Step 3. Name.]  #linebreak() Choose a variable to represent the width. #linebreak() The length is 15 feet more than the width.], [#linebreak() #figure(figph[The image shows the text 'Let W = width' in black characters against a plain white background. The text is centrally aligned and occupies the upper left portion of the frame.], alt: "The image shows the text 'Let W = width' in black characters against a plain white background. The text is centrally aligned and occupies the upper left portion of the frame.", caption: none)

 #linebreak() #figure(figph[A mathematical expression reads 'W + 15 = length' on a white background, indicating a relationship between a variable W, the number 15, and the measurement of length.], alt: "A mathematical expression reads 'W + 15 = length' on a white background, indicating a relationship between a variable W, the number 15, and the measurement of length.", caption: none)],
  [#emph[Step 4. Translate]], [],
  [Write the appropriate formula.], [#figure(figph[The formula P = 2L + 2W is shown, representing the perimeter of a rectangle, where P is perimeter, L is length, and W is width.], alt: "The formula P = 2L + 2W is shown, representing the perimeter of a rectangle, where P is perimeter, L is length, and W is width.", caption: none)],
  [Substitute.], [#figure(figph[A mathematical equation is displayed, reading 150 = 2(W + 15) + 2W. This equation involves the number 150, variables W, and operations of multiplication, addition, and parentheses.], alt: "A mathematical equation is displayed, reading 150 = 2(W + 15) + 2W. This equation involves the number 150, variables W, and operations of multiplication, addition, and parentheses.", caption: none)],
  [#emph[Step 5. Solve]the equation.], [#figure(figph[A mathematical equation shows '150 = 2W + 30 + 2W' in black text on a white background, representing an algebraic problem.], alt: "A mathematical equation shows '150 = 2W + 30 + 2W' in black text on a white background, representing an algebraic problem.", caption: none)

 #linebreak() #figure(figph[A mathematical equation is displayed, reading '150 = 4W + 30' against a plain white background.], alt: "A mathematical equation is displayed, reading '150 = 4W + 30' against a plain white background.", caption: none)

 #linebreak() #figure(figph[A mathematical equation '120 = 4W' is displayed on a white background, suggesting a problem or calculation related to a variable 'W'.], alt: "A mathematical equation '120 = 4W' is displayed on a white background, suggesting a problem or calculation related to a variable 'W'.", caption: none)

 #linebreak() #figure(figph[A mathematical equation shows '30 = W (the width of the pool)', defining the variable W as the width of a pool with a value of 30.], alt: "A mathematical equation shows '30 = W (the width of the pool)', defining the variable W as the width of a pool with a value of 30.", caption: none)

 #linebreak() #figure(figph[The image displays the expression 'W + 15 (the length of the pool)', indicating a mathematical formula for the length of a pool in terms of a variable W.], alt: "The image displays the expression 'W + 15 (the length of the pool)', indicating a mathematical formula for the length of a pool in terms of a variable W.", caption: none)

 #linebreak() #figure(figph[The image displays a simple arithmetic problem, '30 + 15,' written in dark gray and red numbers against a white background.], alt: "The image displays a simple arithmetic problem, '30 + 15,' written in dark gray and red numbers against a white background.", caption: none)

 #linebreak() #figure(figph[The number 45 is displayed in a simple, dark font on a white background, appearing as an isolated numeral.], alt: "The number 45 is displayed in a simple, dark font on a white background, appearing as an isolated numeral.", caption: none)],
  [#emph[Step 6. Check.] #linebreak()  #linebreak() #math.equation(block: false, alt: "P, equals, 2 L plus 2 W; 150, ≟, 2 open parenthesis 45 close parenthesis plus 2 open parenthesis 30 close parenthesis; 150, equals, 150 ✓")[$P & = & 2 L + 2 W \ 150 & ≟ & 2 ( 45 ) + 2 ( 30 ) \ 150 & = & 150 ✓$]], [],
  [#emph[Step 7. Answer]the question.], [The length of the pool is 45 feet and the width is 30 feet.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.

#solutionbox[
70 feet, 30 feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.

#solutionbox[
90 yards, 60 yards
]
]


=== Key Concepts

- #strong[Problem-Solving Strategy for Geometry Applications]
+ #strong[Read] the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for by choosing a variable to represent it.
+ #strong[Translate] into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.
- #strong[Triangle Properties For #math.equation(block: false, alt: "△ A B C")[$"△" A B C$]] #linebreak() 
Angle measures: #linebreak() 
- #math.equation(block: false, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180")[$m "∠" A + m "∠" B + m "∠" C = 180$]


Perimeter: #linebreak() 
- #math.equation(block: false, alt: "P equals a plus b plus c")[$P = a + b + c$]


Area: #linebreak() 
- #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h , b equals base , h equals height")[$A = frac(1, 2) b h , #h(0.2em) "b" = "base" , "h" = "height"$]


A right triangle has one #math.equation(block: false, alt: "90 °")[$90 "°"$] angle.
- #strong[The Pythagorean Theorem] In any right triangle, #math.equation(block: false, alt: "a squared plus b squared equals c squared")[$a^(2) + b^(2) = c^(2)$] where #emph[c] is the length of the hypotenuse and #emph[a] and #emph[b] are the lengths of the legs.
- #strong[Properties of Rectangles]
- Rectangles have four sides and four right (90°) angles.
- The lengths of opposite sides are equal.
- The perimeter of a rectangle is the sum of twice the length and twice the width: #math.equation(block: false, alt: "P equals 2 L plus 2 W .")[$P = 2 L + 2 W .$] The area of a rectangle is the length times the width: #math.equation(block: false, alt: "A equals L W .")[$A = L W .$]


==== Practice Makes Perfect

#strong[Solving Applications Using Triangle Properties]

In the following exercises, solve using triangle properties.

The measures of two angles of a triangle are 26 and 98 degrees. Find the measure of the third angle.

#solutionbox[
56 degrees
]

The measures of two angles of a triangle are 61 and 84 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 105 and 31 degrees. Find the measure of the third angle.

#solutionbox[
44 degrees
]

The measures of two angles of a triangle are 47 and 72 degrees. Find the measure of the third angle.

The perimeter of a triangular pool is 36 yards. The lengths of two sides are 10 yards and 15 yards. How long is the third side?

#solutionbox[
11 feet
]

A triangular courtyard has perimeter 120 meters. The lengths of two sides are 30 meters and 50 meters. How long is the third side?

If a triangle has sides 6 feet and 9 feet and the perimeter is 23 feet, how long is the third side?

#solutionbox[
8 feet
]

If a triangle has sides 14 centimeters and 18 centimeters and the perimeter is 49 centimeters, how long is the third side?

A triangular flag has base one foot and height 1.5 foot. What is its area?

#solutionbox[
0.75 sq. ft.
]

A triangular window has base eight feet and height six feet. What is its area?

What is the base of a triangle with area 207 square inches and height 18 inches?

#solutionbox[
23 inches
]

What is the height of a triangle with area 893 square inches and base 38 inches?

One angle of a right triangle measures 33 degrees. What is the measure of the other small angle?

#solutionbox[
57
]

One angle of a right triangle measures 51 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 22.5 degrees. What is the measure of the other small angle?

#solutionbox[
67.5
]

One angle of a right triangle measures 36.5 degrees. What is the measure of the other small angle?

The perimeter of a triangle is 39 feet. One side of the triangle is one foot longer than the second side. The third side is two feet longer than the second side. Find the length of each side.

#solutionbox[
13 ft., 12 ft., 14 ft.
]

The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.

One side of a triangle is twice the shortest side. The third side is five feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.

#solutionbox[
3 ft., 6 ft., 8 ft.
]

One side of a triangle is three times the shortest side. The third side is three feet more than the shortest side. The perimeter is 13 feet. Find the lengths of all three sides.

The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

#solutionbox[
#math.equation(block: true, alt: "45 ° , 45 ° , 90 °")[$45 "°" , 45 "°" , 90 "°"$]
]

The measure of the smallest angle of a right triangle is 20° less than the measure of the next larger angle. Find the measures of all three angles.

The angles in a triangle are such that one angle is twice the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

#solutionbox[
#math.equation(block: true, alt: "30 ° , 60 ° , 90 °")[$30 "°" , 60 "°" , 90 "°"$]
]

The angles in a triangle are such that one angle is 20° more than the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

#strong[Use the Pythagorean Theorem]

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

#figure(figph[A right triangle with legs marked 9 and 12.], alt: "A right triangle with legs marked 9 and 12.", caption: none)

#solutionbox[
15
]

#figure(figph[A right triangle with legs marked 16 and 12.], alt: "A right triangle with legs marked 16 and 12.", caption: none)

#figure(figph[A right triangle with legs marked 15 and 20.], alt: "A right triangle with legs marked 15 and 20.", caption: none)

#solutionbox[
25
]

#figure(figph[A right triangle with legs marked 5 and 12.], alt: "A right triangle with legs marked 5 and 12.", caption: none)

In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth, if necessary.

#figure(figph[A right triangle with one leg marked 6 and hypotenuse marked 10.], alt: "A right triangle with one leg marked 6 and hypotenuse marked 10.", caption: none)

#solutionbox[
8
]

#figure(figph[A right triangle with one leg marked 8 and hypotenuse marked 17.], alt: "A right triangle with one leg marked 8 and hypotenuse marked 17.", caption: none)

#figure(figph[A right triangle with one leg marked 5 and hypotenuse marked 13.], alt: "A right triangle with one leg marked 5 and hypotenuse marked 13.", caption: none)

#solutionbox[
12
]

#figure(figph[A right triangle with one leg marked 16 and hypotenuse marked 20.], alt: "A right triangle with one leg marked 16 and hypotenuse marked 20.", caption: none)

#figure(figph[A right triangle with one leg marked 8 and hypotenuse marked 13.], alt: "A right triangle with one leg marked 8 and hypotenuse marked 13.", caption: none)

#solutionbox[
10.2
]

#figure(figph[A right triangle with both legs marked 6.], alt: "A right triangle with both legs marked 6.", caption: none)

#figure(figph[A right-angled triangle with a horizontal base of length 5 and a hypotenuse of length 11. The square symbol indicates the right angle between the base and the vertical side.], alt: "A right-angled triangle with a horizontal base of length 5 and a hypotenuse of length 11. The square symbol indicates the right angle between the base and the vertical side.", caption: none)

#solutionbox[
9.8
]

#figure(figph[A right triangle with legs marked 5 and 7.], alt: "A right triangle with legs marked 5 and 7.", caption: none)

In the following exercises, solve using the Pythagorean Theorem. Approximate to the nearest tenth, if necessary.

A 13-foot string of lights will be attached to the top of a 12-foot pole for a holiday display, as shown below. How far from the base of the pole should the end of the string of lights be anchored?

#figure(figph[A right triangle with one leg marked 12 and hypotenuse marked 13.], alt: "A right triangle with one leg marked 12 and hypotenuse marked 13.", caption: none)

#solutionbox[
5 feet
]

Pam wants to put a banner across her garage door, as shown below, to congratulate her son for his college graduation. The garage door is 12 feet high and 16 feet wide. How long should the banner be to fit the garage door?

#figure(figph[A house is shown with a banner over the garage door. The garage door is marked 16 ft wide and 12 ft high.], alt: "A house is shown with a banner over the garage door. The garage door is marked 16 ft wide and 12 ft high.", caption: none)

Chi is planning to put a path of paving stones through her flower garden, as shown below. The flower garden is a square with side 10 feet. What will the length of the path be?

#figure(figph[A square garden is shown that is marked 10’ on the side. There is a path of stones along the diagonal of the square.], alt: "A square garden is shown that is marked 10’ on the side. There is a path of stones along the diagonal of the square.", caption: none)

#solutionbox[
14.1 feet
]

Brian borrowed a 20 foot extension ladder to use when he paints his house. If he sets the base of the ladder 6 feet from the house, as shown below, how far up will the top of the ladder reach?

#figure(figph[A house is shown with a ladder leaning against it. The ladder is marked 20’, and the distance from the house to the base of the ladder is marked 6’.], alt: "A house is shown with a ladder leaning against it. The ladder is marked 20’, and the distance from the house to the base of the ladder is marked 6’.", caption: none)

#strong[Solve Applications Using Rectangle Properties]

In the following exercises, solve using rectangle properties.

The length of a rectangle is 85 feet and the width is 45 feet. What is the perimeter?

#solutionbox[
260 feet
]

The length of a rectangle is 26 inches and the width is 58 inches. What is the perimeter?

A rectangular room is 15 feet wide by 14 feet long. What is its perimeter?

#solutionbox[
58 feet
]

A driveway is in the shape of a rectangle 20 feet wide by 35 feet long. What is its perimeter?

The area of a rectangle is 414 square meters. The length is 18 meters. What is the width?

#solutionbox[
23 meters
]

The area of a rectangle is 782 square centimeters. The width is 17 centimeters. What is the length?

The width of a rectangular window is 24 inches. The area is 624 square inches. What is the length?

#solutionbox[
26 inches
]

The length of a rectangular poster is 28 inches. The area is 1316 square inches. What is the width?

Find the length of a rectangle with perimeter 124 and width 38.

#solutionbox[
24
]

Find the width of a rectangle with perimeter 92 and length 19.

Find the width of a rectangle with perimeter 16.2 and length 3.2.

#solutionbox[
4.9
]

Find the length of a rectangle with perimeter 20.2 and width 7.8.

The length of a rectangle is nine inches more than the width. The perimeter is 46 inches. Find the length and the width.

#solutionbox[
16 in., 7 in.
]

The width of a rectangle is eight inches more than the length. The perimeter is 52 inches. Find the length and the width.

The perimeter of a rectangle is 58 meters. The width of the rectangle is five meters less than the length. Find the length and the width of the rectangle.

#solutionbox[
17 m, 12 m
]

The perimeter of a rectangle is 62 feet. The width is seven feet less than the length. Find the length and the width.

The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.

#solutionbox[
13.5 m length, 12.8 m width
]

The length of the rectangle is 1.1 meters less than the width. The perimeter of a rectangle is 49.4 meters. Find the dimensions of the rectangle.

The perimeter of a rectangle is 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.

#solutionbox[
50 ft., 25 ft.
]

The length of a rectangle is three times the width. The perimeter of the rectangle is 72 feet. Find the length and width of the rectangle.

The length of a rectangle is three meters less than twice the width. The perimeter of the rectangle is 36 meters. Find the dimensions of the rectangle.

#solutionbox[
7 m width, 11 m length
]

The length of a rectangle is five inches more than twice the width. The perimeter is 34 inches. Find the length and width.

The perimeter of a rectangular field is 560 yards. The length is 40 yards more than the width. Find the length and width of the field.

#solutionbox[
160 yd., 120 yd.
]

The perimeter of a rectangular atrium is 160 feet. The length is 16 feet more than the width. Find the length and width of the atrium.

A rectangular parking lot has perimeter 250 feet. The length is five feet more than twice the width. Find the length and width of the parking lot.

#solutionbox[
85 ft., 40 ft.
]

A rectangular rug has perimeter 240 inches. The length is 12 inches more than twice the width. Find the length and width of the rug.


==== Everyday Math

Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are six feet, eight feet and 10 feet. How many feet of fencing will she need to enclose her flowerbed?

#solutionbox[
240 feet
]

Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 50 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 10 feet. How long can he make the other length?


==== Writing Exercises

If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.

#solutionbox[
area; answers will vary
]

If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning.

Look at the two figures below.

#figure(figph[On the left, we have a rectangle with height 2 and width 8. On the right, we have a square with height 4 and width 4.], alt: "On the left, we have a rectangle with height 2 and width 8. On the right, we have a square with height 4 and width 4.", caption: none)

+ ⓐ  Which figure looks like it has the larger area?
+ ⓑ Which looks like it has the larger perimeter?
+ ⓒ Now calculate the area and perimeter of each figure.
+ ⓓ Which has the larger area?
+ ⓔ Which has the larger perimeter?

#solutionbox[
ⓐ Answers will vary. #linebreak() 
ⓑ Answers will vary. #linebreak() 
ⓒ Answers will vary. #linebreak() 
ⓓ The areas are the same. #linebreak() 
ⓔ The 2x8 rectangle has a larger perimeter than the 4x4 square.
]

Write a geometry word problem that relates to your life experience, then solve it and explain all your steps.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “solve applications using triangle properties,” “use the Pythagorean Theorem,” and “solve applications using rectangle properties.” The rest of the cells are blank], alt: "This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “solve applications using triangle properties,” “use the Pythagorean Theorem,” and “solve applications using rectangle properties.” The rest of the cells are blank", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?